One of the of the practice questions from our "Growth Rates Are Crucial" video asks you to compare real GDP per capita for two countries that start at the same

One of the of the practice questions from our "Growth Rates Are Crucial" video asks you to compare real GDP per capita for two countries that start at the same place, but grow at different rates. It’s a little tricky:

Suppose two countries start with the same real GDP per capita, but country A is growing at 2% per year and country B is growing at 3% per year. After 140 years, country B will have a real GDP per capita that is roughly ________ times higher than country A. (Hint- you may want to review the “Rule of 70” to answer this question.)

We asked our Instructional Designer, Mary Clare Peate, to hold virtual “office hours” to guide you through how to solve this problem. Join her as she discusses your questions!

Is there a different practice problem that has you stuck? Suggest a topic for our next office hours in the comments below.

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Today, we're going to answer the following question from our growth-rates video.

 

Suppose two countries start with the same real GDP per capita, but Country A is growing at 2% per year and Country B is growing at 3% per year. After 140 years, Country B's real GDP per capita will be how many times larger than Country A's? Now, before we begin, try to solve this problem on your own. So just put me on pause, slog through this alone and then come back and we'll do this together. And while you're at it, just like turn off your cell phone, close the cat videos. You know the drill.


Okay, ready? The real trick here is to realize that we don't actually need to know this initial value. This number could be 2000; it could be 5 billion. No big deal. As long as these two countries have the same initial starting value, we're fine. We can solve this in three steps. First, find Country A's GDP per capita after 140 years. Next, do the exact same thing for Country B. And finally, compare the two.


Now, for step one we need to find how many times Country A's GDP will double over 140 years. There are a lot of fancy formulas out there. Personally, I prefer the Rule of 70. Sure, it's an approximation. But it makes you feel pretty smart because you can do most of the math in your head. Super simple formula. All it is: 70 divided by the growth rate of the variable equals the time it takes for that variable to double. In the case of Country A, growth rate is 2%. Do the math. 70 divided by 2 is 35. This country is doubling once every 35 years. And over the time horizon of 140 years, this country will double 4 times, and will therefore grow by a factor of 2 to the fourth, or 16.


Now, it's probably right about now you want to call a time-out. Where on earth did you come up with this 2 to the fourth? Honestly, this is a really confusing concept. It tripped me up back in the day as well, so we're going to go over it right now. We know Country A's real GDP per capita will double 4 times over the course of 140 years. But we don't actually know Country A's initial GDP value, so for simplicity let's just call it Y. After 35 years, Y is going to double to 2Y. And after another 35 year, 2Y, Country A's new GDP value, will double yet again to 4Y, and so on. 4Y doubles to 8Y, and 8Y doubles to 16Y at the end of 140 years.


Now, this process of multiplying by 2 every 35 years can just be mathematically simplified to 2 to the fourth. So, our final answer is 2 ^ 4 Y. This isn't exactly expressed in just a normal number like 6,532. Unfortunately, we have to express this final value in terms of A's initial value, Y. We've wrapped up with step one and now we can move on to step two, which is to find Country B's real GDP per capita after 140 years. Now, this is the exact same process we used for A, so it should go pretty quickly. Country B's growth rate is 3%. Plug that in to the rule of 70: 70 / 3 = 23 1/3. So, Country B is doubling once every 23-ish years, such that over 140 years, this country's actually doubling 6 times.


Now let's invoke déjà vu here from step one. Doubling six times is actually the same thing as Country B's initial starting value growing by a factor of 2 to the sixth. And since we actually don't know Country B's initial value, we're going to use Y because we do know it's the same as Country A's initial value. We're now ready to move on to step three and compare them. We can finally return to the initial question we're trying to solve, which is, how many times larger is Country B's GDP per capita than Country A's after 140 years? Now, it's obvious that the two are not equal, but we can set up an equation to compare and solve. In this instance we've added an X, which represents the difference between Country A's GDP and Country B's GDP. And so now we'll just go through and solve for X. Notice immediately we can cancel out those Y's, and then just using the law of exponents in division, we get 4, which means that Country B's GDP per capita is 4 times larger than Country A's after 140 years. And that's the answer.

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